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Digital topological complexity numbers

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The intersection of topological robotics and digital topology leads to us a new workspace. In this paper we introduce the new digital homotopy invariant digital topological complexity number T C(X, κ) for digital images and give some examples and results about it. | Turk J Math (2018) 42: 3173 – 3181 © TÜBİTAK doi:10.3906/mat-1807-101 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Digital topological complexity numbers İsmet KARACA∗,, Melih İS, Department of Mathematics, Faculty of Science, Ege University, İzmir, Turkey Received: 13.07.2018 • Accepted/Published Online: 15.10.2018 • Final Version: 27.11.2018 Abstract: The intersection of topological robotics and digital topology leads to us a new workspace. In this paper we introduce the new digital homotopy invariant digital topological complexity number T C(X, κ) for digital images and give some examples and results about it. Moreover, we examine adjacency relations in the digital spaces and observe how T C(X, κ) changes when we take a different adjacency relation in the digital spaces. Key words: Topological complexity number, digital topology, digital topological complexity number 1. Introduction Topological robotics is an area that uses some topological properties to compute the homotopy invariant topological complexity number T C(X) , mostly in mechanics and robot motion planning problems. Michael Farber [16] first introduced the topological complexity number T C(X) that measures the degree of the deflection of the contractibility of a given area for a robot . After that, Farber published many works on this subject, such as [17–20]. Besides Farber, Grant [23], Dranishnikov [12], Tabachnikov and Yuzvinsky [22], and other researchers have improved it with their different works. Simply, T C(X) computes the minimum number that is an invariant in topology (and of course in algebraic topology too) for a robot to move from our given initial point to our given final point. This number requires a continuous motion planning algorithm. A motion planning algorithm in a path-connected space X takes two points a and b of X and composes a path α in X such that α(0) = a and α(1) = b. Therefore, we observe that robotics are motivated by this